Magnetic Domains in Spinor Bose–Einstein Condensates

Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9, p. 883–890

Magnetic domains in spinor Bose–Einstein condensates

Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar

Nonlinear Physics Center and ARC Center of Excellence for Quantum-Atom Optics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia E-mail: [email protected]

Received December 14, 2009

We discuss the structure of spin-1 Bose–Einstein condensates in the presence of a homogenous magnetic field. We demonstrate that the phase separation can occur in the ground state of antiferromagnetic (polar) condensates, while the spin components of the ferromagnetic condensates are always miscible, and no phase separation occurs. Our analysis predicts that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. We propose a method for generation of spin domains by adiabatic switching of the magnetic field. We also discuss the phenomena of dynamical instability and spin domain formation.

PACS: 03.75.Lm Tunneling, Josephson effect, Bose–Einstein condensates in periodic potentials, solitons, vortices, and topological excitations; 05.45.Yv Solitons.

Keywords: Bose–Einstein condensates, homogeneous magnetic field, magnetic domains.

1. Introduction A spin-1 BEC in a magnetic field is subjected to the well-known Zeeman effect. At low fields the effect is dom- The analysis of the properties of spin domains and inated by the linear Zeeman effect, which leads to a Lar- magnetic solitons is one of the major topics of the theory mor precession of the spin vector about the magnetic field of crystalline magnetic structures [1]. However, the recent at a constant rate, which is unaffected by spatial inhomo- development in the physics of cold gases gave a birth to an geneities in the condensate [10]. At higher magnetic fields exciting new field where the spinor dynamics and magnet- the quadratic Zeeman effect becomes important, and leads ic domain formation are the key ingredients of a new and to much more dramatic effects in the condensate, such as seemingly different physics which, however, borrows coherent population exchange between spin components many important results and techniques from the solid state [8,11,12] and the breaking of the single-mode approxima- physics. More specifically, the spin degree of freedom of tion (SMA) [13,14], which assumes that all the spin com- spinor Bose–Einstein condensates (BECs) [2–4] leads to a ponents share the same spatial density and phase profile. wealth of new phenomena not possessed by single- The study of the behavior of a spin-1 condensate in the component (spin-frozen) condensates. New spin-induced presence of a magnetic field began with the work of Sten- dynamics such as spin waves [3], spin-mixing [5] and spin ger et al. [2], where the existence of magnetic (spin) do- textures [3,6] have all been predicted theoretically and ob- mains were predicted and observed in the ground state of a served in experiment. The observation of these spin- polar 23Na condensate subject to a magnetic field gradient. dependent phenomena became possible due to the devel- At the same time, the ground states of both ferromagnetic opment of optical traps [7] which trap all spin components, and antiferromagnetic (polar) condensates in homogenous rather than just the low-magnetic-field seeking spin states magnetic field were found to be free of spin domains in the of magnetic traps. However, the effect of an additional local density approximation. It was later found that the small non-zero magnetic field on the condensate in these SMA was broken in the ground state of a condensate con- optical traps was studied even in the seminal theoretical [4] fined in a harmonic trap even in a homogenous field [13]. and experimental [2] works. In fact the interplay of spin Nevertheless, the SMA continued to be used in studies of and magnetic field has been at the heart of some of the spinor condensates for its simplicity and validity in a broad most impressive spinor BEC experiments, including the range of experimental situations [11,15], in particular when demonstration of spin domains [2], spin oscillations [8] the condensate size is smaller than the spin healing length, and observation of spin textures and vortices [9]. which determines the minimum domain size. On the other

© Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar, 2010 Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar hand, the dynamical instability, leading to the spontaneous atoms with total spin S . The total number of atoms and formation of dynamic spin domains, was found to occur in the total magnetization large ferromagnetic condensates prepared in excited initial Nnd=,r (4) states [9,16–18], while no such phenomenon was predicted ∫ to occur [16] or observed [19] in antiferromagnetic con- M ==F dnndrr , densates. Similar instabilities were found in the transport ∫∫z ()+−− (5) of both types of spin-1 condensates in optical lattices [20]. are conserved quantities. The Zeeman energy shift for each It seemed however that spin domains were only to be found of the components, E can be calculated using the Breit– in antiferromagnetic condensates in the presence of inho- j Rabi formula [26] mogeneous magnetic fields [2] or trapping potentials. In this work, first we overview the results of our recent 1 ⎛⎞2 EE± =141,−HFS⎜⎟ + ±α+αm gB I μ B works [21,22] and discuss the structure of spin-1 Bose– 8 ⎝⎠ Einstein condensates in the presence of a homogenous 1 ⎛⎞2 magnetic field. We show that the translational symmetry of EE=141,−++α⎜⎟ (6) 0 8 HFS ⎝⎠ a homogenous BEC is spontaneously broken and phase separation occurs in magnetized polar condensates if the where EHFS is the hyperfine energy splitting at zero mag- magnetic field is strong enough. An analogous phenome- netic field, α =(ggI +μJB ) BE / HFS, where μB is the non has been predicted and observed previously in binary Bohr magneton, gI and gJ are the gyromagnetic ratios of condensates [23,24]. In contrast, the cases when domain electron and nucleus, and B is the magnetic field strength. formation is driven by inhomogeneous external potentials The linear part of the Zeeman effect gives rise to an overall or magnetic field gradients may be referred to as potential shift of the energy, and so we can remove it with the trans- separation according to the naming used in Ref. 24. Here, formation we show that for a range of experimental conditions, it is HHN→+( +MM ) E /2 + ( N − ) E /2. (7) energetically favorable for the system to consist of two +− separate phases composed of different stationary states. This transformation is equivalent to the removal of the Finally, we demonstrate numerically that this phenomenon Larmor precession of the spin vector around the z-axis can be observed in a polar condensate trapped in a harmon- [21]. We thus consider only the effects of the quadratic ic optical potential. Zeeman shift. For sufficiently weak magnetic field we can 2 approximate it by δ+−≈αEEE=(+− 2 E0 )/2 EHFS /16, 2. Model which is always positive. We consider dilute spin-1 BEC in a homogenous mag- The asymmetric part of the Hamiltonian (2) can now be netic field pointing in the z -direction. The mean-field rewritten as Hamiltonian of this system is given by the expression, ⎛⎞c2 2 HdEna =||=(),rFrr⎜⎟−δ0 + dne (8) ∫∫⎝⎠2 ⎛⎞− 2 c H =||()||,dnVHrr⎜⎟h ∇ψ*22 ∇ψ +0 ψ + ψ + ∫ ∑ ⎜⎟j jj jawhere the energy per atom e()r is given by [11] j=,0,−+⎝⎠22M cn cn (1) eE=||=||,−δ ρ +22ff222 −δ E ρ + + m 0022()⊥ where ψψψ−+,,0 are the wavefunctions of atoms in magnetic sub-levels m =1,0,1−+, M is the atomic mass, |f |222 = 2ρ (1−ρ ) + 2 ρ (1 −ρ ) −m cos θ . (9) 2 ⊥ 00 0 0 V ()r is an external potential and nn==||∑∑j ψ j is the total atom density. The asymmetric part of the Hamil- We express the wavefunctions as ψρj =exp()nijj θ tonian is presented as: where the relative densities are ρ jj=/nn. We also intro- duced the relative phase θ =2θ+θ−θ+− 0 , spin per atom ⎛⎞c fF=/n , and magnetization per atom mf==ρ −ρ . Hd=||,rF⎜⎟ En+ 2 2 (2) z + − ajj∫ ⎜⎟∑ 2 The perpendicular spin component per atom is ⎝⎠j=,0,−+ 222 ||=f⊥ fx + f y . where E j is the Zeeman energy shift for state ψ j and the The Hamiltonian (1) generates the Gross–Pitaevskii eq- spin density is, uations describing the mean-field dynamics of the system †††ˆˆˆ ∂ψ± 2* F =(FFFxyz , , )=(ψψψψψψ F x , F y , F z ), (3) icnnnch =([]L + 20±±+− )ψ + 20ψψ , (10) ∂t mm where Fˆ are the spin matrices [25] and ψ = x,,yz ∂ψ0 * iEcnnch =()2,[]L − δ+2020+− + ψ + ψψψ +− =ψ(,,)+− ψ0 ψ . The nonlinear coefficients are given by ∂t 2 2 caaM020=4π+h (2 )/3 and caaM220=4π−h ( )/3 , where L is given by L =/2()−∇22 McnV + + r . where aS is the s-wave scattering length for colliding h 0

884 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 Magnetic domains in spinor Bose–Einstein condensates

By comparing the kinetic energy with the interaction |m|– 1– |m| – 200 energy, we can define a characteristic healing length u. 150

,a.u. , 100 ξπ=2 / 2Mcn0 and spin healing length ξπs =2 / 2Mcn2 . r r j h h a n c ± 0 50

These quantities give the length scales of spatial variations n, n 0 in the condensate profile induced by the spin-independent –200 –100 0 100 200 or spin-dependent interactions, respectively. Analogously, x, mm |m|– |m |– |m| – 200 we define magnetic healing length as ξπ=2 / 2M δE . 2c B h u. 150

In real spinor condensates, the a0 and a2 scattering a. 100

,a.u. , b 2C r0 j d lengths have similar magnitude. The spin-dependent inte- n 50

n, n raction coefficient c2 is therefore much smaller than its 0 spin-independent counterpart c . For example, this ratio is –200 –100 0 100 200 0 x, mm about 1:30 in a 23Na condensate and 1:220 in a 87Rb condensate far from Feshbach resonances [27]. As a result, the Fig. 1. Schematic structure of the phase separated states ρ +ρ excitations that change the total density require much more ± 0 (a) and 2C + ρ (b). The shaded region, in which all three com- energy than those that keep n()r close to the ground state 0 ponents are nonzero, has the approximate extent of one spin heal- profile. In our considerations we will assume that the ing length ξ or magnetic healing length ξ , whichever is amount of energy present in the system is not sufficient to s B greater. The relative size of the domains is indicated with arrows. excite the high-energy modes, and we will treat the total The corresponding wavefunction profiles obtained numerically atom density n()r as a constant. 23 with periodic boundary conditions in the case of Na for m =0.5 with δEcn/( )=0.8 (c) and δEcn/ ( ) = 0.23 (d). The 3. Condensate without a trapping potential 2 2 n+ , n0 , and n− components are depicted by dash-dotted, da- The ground states of spin-1 condensates in homogenous shed, and dotted lines, respectively. The solid lines show the total magnetic field have been studied in a number of previous density. works [2,13,28]. The most common procedure [2] involves minimization of the energy functional with constraints on We construct ground states of the condensate using the number of atoms N and the total magnetization M. homogeneous stationary solutions of the GP equations (10) The resulting Lagrange multipliers p and q serve as pa- −μ+μiti()j Sj +θ rameters related to the quadratic Zeeman shift δE and the ψ jj(,)=r tn e , (11) magnetization m . An alternative method, elaborated in [13], consists of minimization of the energy functional in where μS =/cn0 h is a constant and μ+μ+−=2 μ0 due the parameter space of physically relevant variables B and to a phase matching condition. Following [22], we distin- m . Most of the previous studies, however, were assuming guish several types of stationary states. The states where that the condensate remains homogenous and well de- only a single Zeeman component is populated ( n j =1 for scribed by the single-mode approximation; in particular, a specified j =,0,− or + ) are named ρ− , ρ0 , and ρ+ , the spatial structure observed in [2] resulted from the ap- respectively. The state where n0 =0 but nn−+,0≠ is the plied magnetic field gradient, but the BEC was assumed to two-component (2C) state. The three-component states are be well described by the homogenous model at each point classified according to the value of θθ+θ−θ=2+− 0 . The in space (local density approximation). In Ref. 13, the states with θ =0 are called phase-matched (PM) states, breakdown of the single-mode approximation was shown and the ones with θ = π are called anti-phase-matched numerically for a condensate confined in a harmonic po- (APM) states. For more details about this classification and tential. the properties of the stationary states, refer to [22]. We correct the previous studies by showing that when Two types of domain structures, depicted in Fig. 1, are the condensate size is larger than the spin healing length composed of two different stationary states connected with ξs , the translational symmetry is spontaneously broken a shaded region where all three components are nonzero. and phase separation occurs in magnetized polar conden- These two domain states have the advantage that the per- sates if the magnetic field is strong enough. This pheno- pendicular spin is nonzero only in the transitory region, menon takes place when the energy of the spin state with hence their energy is relatively low in polar condensates. the lowest energy is a concave function of m for a given In fact, these are the only phase separated states that can be δE . On the contrary, the energy is always a convex func- the ground states of a homogenous condensate. Their ener- tion of m for the ferromagnetic condensate, and no phase gies per atom in the limit of infinite condensate size, which separation occurs. Note that phase separation has been pre- allows for neglecting of the relatively small intermediate viously predicted in binary condensates [23,24] and in fer- region are romagnetic condensates at finite temperature [29]. ememe=| |+− (1 | |) , ρ±±+ρ00 ρ ρ

Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 885 Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar

mm⎛⎞ 1.0 ee=|1+− e , (12) 22=CCmm+ρ020C ⎜⎟ ρ mm22CC⎝⎠ |m|– where mN=/M is the average magnetization and the magnetization of the 2C component m2C is a free parameter that has to be optimized to obtain the lowest energy 0.5 state.

Table 1. Ground states of spin-1 condensates in homogenous magnetic field. The states 2C +ρ0 and ρ+ρ± 0 correspond to phase separation (see Fig. 1)

Condensate Parameter range Ground state 0 0.5 1.0 δE r 2 ≤ and m =0 dE/c n 0 ρ0 2 ||cn2 Ferromagnetic Fig. 2. Ground state phase diagram of the polar condensate. The δE <2 or m ≠ 0 PM symbols correspond to numerical data obtained for the parameters ||cn 2 of 23Na, with solid triangles representing 2C, open circles 2C+ m =0 ρ 0 +ρ0 and open squares ρ± +ρ0 . The solid lines and shading are 2 δE m given by the analytical formulas from Table 1. ≤ 2C cn2 2 Polar 2 ∂ψ% 0 * mEδ 1 iEcnnch =()2,⎡⎤⎣⎦L% − δ+%%2020+− + %ψ%%%%+ %ψψψ +− << and m ≠ 0 2C + ρ0 ∂t 22cn2 222 1 δE with L%=(− /2)mxc∂∂ / +%0 , where caa%020=4ω+⊥ (2 )/3, ≤ and m ≠ 0 h h ρ± + ρ0 2 cn2 caa%220=4hω⊥ (− )/3, ∫ dx∑ |ψ% j | = N , and ω⊥ is the transverse trapping frequency. We imposed periodic boun- The ground states can be determined by comparing dary conditions on ψ% j ()x and used the parameters corres- energies of the phase separated states with the energies of ponding to a 23Na BEC containing N =5.2·104 atoms con- 3 the homogenous solutions. The form of formulas (12) indi- fined in a transverse trap with frequency ωπ⊥ =2 ·10 . The cates that the phase separation will occur when the energy Fermi radius of the transverse trapping potential is smaller of the lowest homogenous state is a concave function of than the spin healing length, and the nonlinear energy scale magnetization. The results for both polar and ferromagnet- is much smaller than the transverse trap energy scale, ic condensates are collected in Table 1. In the cases when which allows us to reduce the problem to one spatial di- no phase separation occurs, our results are in agreement mension [27,31]. The solutions were found numerically with those obtained in [13]. Note that we assumed that the using the normalized gradient flow method [32,33], which condensate size is much larger than ξs and ξB . For small is able to find a state which minimizes the total energy for condensates, the results of [13] are correct. given N and M, and fulfills the phase matching condi- In the case of high magnetic field strength, one of the tion. The stability of the resulting states was verified using Zeeman sub-levels is practically depleted [13] and the numerical time evolution according to Eqs. (13). The slight condensate becomes effectively two-component. The exis- discrepancy between numerical and analytical results can tence of the ρ± + ρ0 phase in a polar condensate can then be accounted for by the finite size of the condensate (the be understood within the binary condensate model [23,24]. box size was ∼10ξs ), and by the deviation from the as- We note that the experiment reported in Ref. 30, performed sumption that the total density is constant (see the discus- in this regime, can be viewed as the first confirmation of sion at the end of Sec. 2). Due to the finite value of the phase separation in spin-1 BEC in a homogenous magnetic ratio cc20/ there is a slight density modulation, as is evi- field. However, the ground state was not achieved, and a dent in Fig. 1,c,d. multiple domain structure was observed. In Fig. 2 we present the phase diagram of polar conden- 4. Condensate trapped in a harmonic optical potential sates, obtained both numerically and using analytical formulas from Table 1. The ground state profiles for a quasi- The results from the preceding subsection can be veri- 1D condensate were found numerically by solving the 1D fied experimentally in configurations involving toroidal or version of Eqs. (10) [21] square-shaped optical traps [34]. However, in most experiments on BECs, harmonic potentials are used. The relev- ∂ψ% ± 2* icnnnch =(⎡⎤L% ++−%%20±± % % )ψ%%%+ % 20ψψ , (13) ance of these results is not obvious in the case of harmonic ∂t ⎣⎦mm trapping, since the coefficient δEcn/(2 ), one of the main

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200 200 a b

150 150

u. u.

a. a.

,a.u. , 100 ,a.u. , 100

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0 0 –200 –100 0 100 200 –200 –100 0 100 200 x, µm x, µm 200 200 c d 150 150

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a. a.

,a.u. , 100 ,a.u. , 100

j j

n n

n, n n, n 50 50

0 0 –200 –100 0 100 200 –200 –100 0 100 200 x, µm x, µm

Fig. 3. Ground state profiles in a harmonic trap potential. Phase separation occurs in the polar 23 Na condensate when the magnetic field strength is increased from B =0.1G, δEcn/(2max )=0.09 (a) to B =0.12G, δEcn/(2max )=0.13 (b) and B =0.25G, 87 δEcn/(2max )=0.56 (c). For comparison, the ground state of a Rb condensate is shown in for B =0.2G, δ−Ecn/(2max )= 0.41 (d). The n+ , n0 , and n− components are depicted by dash-dotted, dashed, and dotted lines, respectively. The solid lines shows the total 4 23 87 3 density. Other parameters are N =2.1·10 , ωπ =2 ·10 ( Na), ω =2π ·7.6 ( Rb), ωπ⊥ =2 ·10 and m =0.5. parameters controlling the condensate properties, varies in be explained by calculating the total asymmetric energy of space due to the varying total density n . the condensate (8), again assuming that the contribution The ground states in a highly elongated harmonic trap, from the intermediate region connecting the domains is where the parallel part of the potential has the form negligible, Vx()=(1/2) Mω22 x, are presented in Fig. 3 [33]. We can cn see that as the magnetic field strength is increased, phase HdnEdn≈ rr()−δ +2 = a ∫∫2 separation occurs and the ρ± + ρ0 domain state is formed. ρρ0 ± However, in contrast to the previous case, the transition is cN2 not sharp, and in particular there is no distinct 2C +ρ =(||)−δEN −M + 〈 n 〉ρ , (14) 0 2 ± phase for any value of the magnetic field. Note that the where 〈n〉 is the mean condensate density within the state in Fig. 3,a is also spatially separated due to different ρ± area of the ρ domain. We see that the energy will be the Thomas–Fermi radii of the ψ− and ψ+ components; ± however, this is an example of potential separation, as op- lowest if this domain is localized in the outer regions, posed to phase separation [24], since it is does not occur in where the condensate density is low. the absence of the potential. On the other hand, Fig. 3,d shows that the components of ferromagnetic condensate 5. Generation of spin domains are miscible even in the regime of strong magnetic field. In We propose a method for generation of spin domains the regions where the wavefunctions overlap, the relative described in the previous section by adiabatic switching of phase is equal to θ =0 for ferromagnetic and θ = π for the magnetic field. We start with a condensate with all the polar ground states, since these configurations minimize atoms in the m =1 sublevel, in the ground state of a har- the spin energy (9). monic potential. Subsequently, some of the atoms are The characteristic feature of phase separation in the po- transferred to the m =1− component in the rapid adiabatic lar BEC is that the m =0 domain tends to be localized in passage process. The magnetic field is then suddenly the center of the trap, as shown in Fig. 1,b and c. This can switched off. In this way we can obtain a condensate with

Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 887 Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar

200 a d 100 150

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0 0 200 400 600 800 1000 –200 –100 0 100 200 t,ms x, µm 200 b e 100 150

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µm 0

,a.u. ,

j 100

x, µm

n, n –100 50

0 100 200 300 400 500 0 –200 –100 0 100 200 t,ms x, µm

Fig. 4. Generation of spin domains by switching the magnetic field on. The magnetic field is gradually increased from zero to final value B =0.15G during t =1s (a, d), B =0.25G during t =2s (b, e), and B =0.25G during t =0.5s (c, f). The left column shows the time dependence of the atom density in the initially unoccupied m =0 component, and the right column shows the final domain profiles. In the last case, corresponding to nonadiabatic switching, multiple domains are formed. Other parameters are the same as in Fig. 3. arbitrary magnetization in the 2C state, which is the ground shows the time dependence of the atom density in the in- state in B =0, see Fig. 2. Next, we gradually increase the itially unoccupied m =0 component, and the right column magnetic field strength in an adiabatic process according to shows the final domain profiles. These should be compared the formula with the ground state profiles in Fig. 3. The domains are generated for both the low and high magnetic field cases in t times of the order of seconds, as shown in panels (a, d) and BB= final (15) tswitch (b, e). However, when the switching time is significantly reduced, yielding the process no longer adiabatic, multiple where t =0 at the beginning of the switching process, metastable domains are formed as presented in panels t is the switching time, and B is the desired final switch final (c, f). This picture is in qualitative agreement with the ex- value of the magnetic field. The form of Eq. (15) assures periment [30], where metastable spin domains were that the quadratic Zeeman splitting grows linearly in time. formed in nonadiabatic process within 50–100 ms. We have confirmed that this condition improves the adia- baticity of the generation process. We present examples of the evolution of the condensate in Fig. 4. The left column

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6. Spin domains and dynamical stability latter case the instability growth rate of unstable modes is proportional to the fourth power of the magnetic field Our results presented above show that the domain struc- strength. The time required for the development of instabil- ture forming in polar condensates is absent in ferromagnet- ity may be much longer than the condensate lifetime [3]. ic BECs. This may seem to contradict the common under- standing of ferromagnetism and the results of the quenched 7. Conclusions BEC experiment Ref. 9. The conventional picture of a ferromagnet involves many domains pointing in various di- We have studied the ground state of a spin-1 BEC in the rections separated by domain walls. Similar structure has presence of a homogenous magnetic field with and without been observed in Ref. 9. However, these cases correspond an external trapping potential. We have found that without to the situations when there is an excess kinetic energy a trapping potential the translational symmetry can be present in the system, due to finite temperature or excita- spontaneously broken in polar BEC, with the formation of tion of the spatial modes. On the other hand, our study is magnetic domains in the ground state. We have shown that limited to the ground states at T =0. It is easy to see from these results may be used to understand the ground state Eq. (9) that in zero magnetic field the ground state of a structure in the presence of a trapping potential by map- ferromagnetic BEC will always consist of a single domain ping the locally varying density in the trap to the homo- with maximum possible value of the spin vector ||=1f , genous state. We have found that, depending on the mag- pointing in the same direction at all points in space. How- netic field, the antiferromagnetic BEC ground state in the ever, when the temperature is finite, more domains can be trap displays pronounced spin domains for a range of poss- formed each with a different direction of the spin vector. ible experimental conditions. Finally, we have discussed We emphasize that the domain structure of the ground the relationship between the phenomenon of phase separa- state in polar condensates is very different from the do- tion and the dynamical instability leading to the formation mains formed when the kinetic energy is injected in the of dynamic spin textures. system as in Ref. 9. The latter constantly appear and disap- pear in a random sequence [9,16,17,21,35,36]. On the con- Acknowledgments trary, the ground state domains are stationary and are posi- This work was supported by the Australian Research tioned in the center of the trap. They exist in the lowest- Council through the ARC Discovery Project and Center of energy state, while the dynamical domains require an Excellence for Quantum-Atom Optics. amount of kinetic energy to be formed. The ground state domains can be prepared in an adiabatic process, involving 1. V.G. Baryakhtar, M.V. Chetkin, B.A. Ivanov, and S.N. Ga- adiabatic rf sweep or a slow change of the magnetic field detskii, Dynamics of Topological Magnetic Solitons: Experi- [30,35], while the kinetic domains require a sudden quench ment and Theory, Springer-Verlag, Heidelberg (1994); and [9,35]. references therein. The dynamical instability of ferromagnetic condensates 2. J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.-J. Miesner, that leads to spontaneous formation of spin domains has A.P. Chikkatur, and W. Ketterle, Nature (London) 396, 345 been investigated theoretically [18,16,35] and observed in (1998). experiment [9]. An analogous phenomenon has been pre- 3. T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). dicted recently for polar condensates in presence of mag- 4. T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). netic field [21]. Here we correct the results of Ref. 21, by 5. M.-S. Chang, C.D. Hamley, M.D. Barrett, J. A. Sauer, K.M. noting that the ρ0 =1 state is stable in ferromagnetic con- Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. densates for δEcn>2|2 | , and the 2C ( ρ0 =0) state is Lett. 92, 140403 (2004). 2 stable in polar BECs if δEm

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12. J. Kronjäger, C. Becker, M. Brinkmann, R. Walser, P. Na- 26. S. Wüster, T. E. Argue, and C.M. Savage, Phys. Rev. A72, vez, K. Bongs, and K. Sengstock, Phys. Rev. A72, 063619 043616 (2005). (2005). 27. B.J. Dąbrowska-Wüster, E.A. Ostrovskaya, T.J. Alexander, 13. W.X. Zhang, S. Yi, and L. You, New J. Phys. 5, 77 (2003). and Y.S. Kivshar, Phys. Rev. A75, 023617 (2007). 14. S. Yi, Ö.E. Müstecaplioglu, C.P. Sun, and L. You, Phys. 28. F. Zhou, Phys. Rev. Lett. 87, 080401 (2001). Rev. A66, 011601(R) (2002). 29. T. Isoshima, T. Ohmi, and K. Machida, J. Phys. Soc. Jpn. 69, 15. D.R. Romano and E.J.V. de Passos, Phys. Rev. A70, 043614 3864 (2000). (2004). 30. H.-J. Miesner, D.M. Stamper-Kurn, J. Stenger, S. Inouye, 16. W. Zhang, D.L. Zhou, M.-S. Chang, M.S. Chapman, and L. A.P. Chikkatur, and W. Ketterle, Phys. Rev. Lett. 82, 2228 You, Phys. Rev. Lett. 95, 180403 (2005). (1999). 17. H. Saito, and M. Ueda, Phys. Rev. A72, 023610 (2005). 31. L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A65, 18. N.P. Robins, W. Zhang, E.A. Ostrovskaya, and Y.S. Kiv- 043614 (2002); W. Zhang and L. You, Phys. Rev. A71, shar, Phys. Rev. A64, 021601(R) (2001). 025603 (2005). 19. A.T. Black, E. Gomez, L.D. Turner, S. Jung, and P.D. Lett, 32. W. Bao and F.Y. Lim, SIAM J. Sci. Comput. 30, 1925 Phys. Rev. Lett. 99, 070403 (2007). (2008). 20. J. Rustekoski and Z. Dutton, Phys. Rev. A76, 063607 (2007). 33. F.Y. Lim and W. Bao, Phys. Rev. E78, 066704 (2008). 21. M. Matuszewski, T.J. Alexander, and Yu.S. Kivshar, Phys. 34. E.M. Wright, J. Arlt, and K. Dholakia, Phys. Rev. A63, Rev. A78, 023632 (2008). 013608 (2000); K.E. Strecker, G.B. Partridge, A.G. Truscott, 22. M. Matuszewski, T.J. Alexander, and Yu. S. Kivshar, Phys. and R.G. Hulet, Nature 417, 150 (2002); S.K. Schnelle, E.D. Rev. A80, 023602 (2009). van Ooijen, M.J. Davis, N.R. Heckenberg, and H. Rubinsz- 23. T.-L. Ho and V.B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996); tein-Dunlop, Opt. Express 16, 1405 (2008). H. Pu and N.P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998); 35. H. Saito, Y. Kawaguchi, and M. Ueda Phys. Rev. A75, M. Trippenbach, K. Goral, K. Rzazewski, B. Malomed, and 013621 (2007). Y.B. Band, J. Phys. B33, 4017 (2000). 36. J. Mur-Petit, M. Guilleumas, A. Polls, A. Sanpera, M. Le- 24. E. Timmermans, Phys. Rev. Lett. 81, 5718 (1998). wenstein, K. Bongs, and K. Sengstock, Phys. Rev. A73, 25. T. Isoshima, K. Machida, and T. Ohmi, Phys. Rev. A60, 013629 (2006). 4857 (1999).

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